Integrand size = 30, antiderivative size = 425 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=-\frac {1}{504} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right ) \sqrt {a+b x^4}-\frac {2 b c \sqrt {a+b x^4}}{45 a x^5}-\frac {b d \sqrt {a+b x^4}}{16 a x^4}-\frac {2 b e \sqrt {a+b x^4}}{21 a x^3}-\frac {b f \sqrt {a+b x^4}}{6 a x^2}+\frac {2 b^2 c \sqrt {a+b x^4}}{15 a^2 x}-\frac {2 b^{5/2} c x \sqrt {a+b x^4}}{15 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b^2 d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 a^{3/2}}+\frac {2 b^{9/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{7/4} \left (7 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 a^{7/4} \sqrt {a+b x^4}} \]
1/16*b^2*d*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)-1/504*(56*c/x^9+63*d/x ^8+72*e/x^7+84*f/x^6)*(b*x^4+a)^(1/2)-2/45*b*c*(b*x^4+a)^(1/2)/a/x^5-1/16* b*d*(b*x^4+a)^(1/2)/a/x^4-2/21*b*e*(b*x^4+a)^(1/2)/a/x^3-1/6*b*f*(b*x^4+a) ^(1/2)/a/x^2+2/15*b^2*c*(b*x^4+a)^(1/2)/a^2/x-2/15*b^(5/2)*c*x*(b*x^4+a)^( 1/2)/a^2/(a^(1/2)+x^2*b^(1/2))+2/15*b^(9/4)*c*(cos(2*arctan(b^(1/4)*x/a^(1 /4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^( 1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^ 2*b^(1/2))^2)^(1/2)/a^(7/4)/(b*x^4+a)^(1/2)-1/105*b^(7/4)*(cos(2*arctan(b^ (1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin( 2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(5*e*a^(1/2)+7*c*b^(1/2))*(a^(1/ 2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(7/4)/(b*x^4+a )^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.35 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=-\frac {\sqrt {a+b x^4} \left (14 a^3 c \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {1}{2},-\frac {5}{4},-\frac {b x^4}{a}\right )+3 x^2 \left (6 a^3 e \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{2},-\frac {3}{4},-\frac {b x^4}{a}\right )+7 x \left (a+b x^4\right ) \sqrt {1+\frac {b x^4}{a}} \left (a^2 f+b^2 d x^6 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},3,\frac {5}{2},1+\frac {b x^4}{a}\right )\right )\right )\right )}{126 a^3 x^9 \sqrt {1+\frac {b x^4}{a}}} \]
-1/126*(Sqrt[a + b*x^4]*(14*a^3*c*Hypergeometric2F1[-9/4, -1/2, -5/4, -((b *x^4)/a)] + 3*x^2*(6*a^3*e*Hypergeometric2F1[-7/4, -1/2, -3/4, -((b*x^4)/a )] + 7*x*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*(a^2*f + b^2*d*x^6*Hypergeometric 2F1[3/2, 3, 5/2, 1 + (b*x^4)/a]))))/(a^3*x^9*Sqrt[1 + (b*x^4)/a])
Time = 0.74 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2364, 27, 2372, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^4} \left (c+d x+e x^2+f x^3\right )}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 2364 |
| \(\displaystyle -2 b \int -\frac {84 f x^3+72 e x^2+63 d x+56 c}{504 x^6 \sqrt {b x^4+a}}dx-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{252} b \int \frac {84 f x^3+72 e x^2+63 d x+56 c}{x^6 \sqrt {b x^4+a}}dx-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 2372 |
| \(\displaystyle \frac {1}{252} b \int \left (\frac {72 e x^2+56 c}{x^6 \sqrt {b x^4+a}}+\frac {84 f x^2+63 d}{x^5 \sqrt {b x^4+a}}\right )dx-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{252} b \left (-\frac {12 b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+7 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}+\frac {168 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}+\frac {63 b d \text {arctanh}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {168 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {168 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {56 c \sqrt {a+b x^4}}{5 a x^5}-\frac {63 d \sqrt {a+b x^4}}{4 a x^4}-\frac {24 e \sqrt {a+b x^4}}{a x^3}-\frac {42 f \sqrt {a+b x^4}}{a x^2}\right )-\frac {1}{504} \sqrt {a+b x^4} \left (\frac {56 c}{x^9}+\frac {63 d}{x^8}+\frac {72 e}{x^7}+\frac {84 f}{x^6}\right )\) |
-1/504*(((56*c)/x^9 + (63*d)/x^8 + (72*e)/x^7 + (84*f)/x^6)*Sqrt[a + b*x^4 ]) + (b*((-56*c*Sqrt[a + b*x^4])/(5*a*x^5) - (63*d*Sqrt[a + b*x^4])/(4*a*x ^4) - (24*e*Sqrt[a + b*x^4])/(a*x^3) - (42*f*Sqrt[a + b*x^4])/(a*x^2) + (1 68*b*c*Sqrt[a + b*x^4])/(5*a^2*x) - (168*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*a ^2*(Sqrt[a] + Sqrt[b]*x^2)) + (63*b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4 *a^(3/2)) + (168*b^(5/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[ a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*a^( 7/4)*Sqrt[a + b*x^4]) - (12*b^(3/4)*(7*Sqrt[b]*c + 5*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcT an[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*a^(7/4)*Sqrt[a + b*x^4])))/252
3.6.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a + b*x^n)^p, x] - Simp[b*n*p Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a, b} , x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1 , 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 ] && !PolyQ[Pq, x^(n/2)]
Result contains complex when optimal does not.
Time = 2.62 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-672 b^{2} c \,x^{8}+840 a b f \,x^{7}+480 a e b \,x^{6}+315 x^{5} d b a +224 a b c \,x^{4}+840 a^{2} f \,x^{3}+720 a^{2} e \,x^{2}+630 a^{2} d x +560 a^{2} c \right )}{5040 x^{9} a^{2}}-\frac {b^{2} \left (\frac {80 a e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {112 i \sqrt {b}\, c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {105 \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )}{840 a^{2}}\) | \(299\) |
| elliptic | \(-\frac {c \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {d \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {e \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {f \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b c \sqrt {b \,x^{4}+a}}{45 a \,x^{5}}-\frac {b d \sqrt {b \,x^{4}+a}}{16 a \,x^{4}}-\frac {2 b e \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {b f \sqrt {b \,x^{4}+a}}{6 a \,x^{2}}+\frac {2 b^{2} c \sqrt {b \,x^{4}+a}}{15 a^{2} x}-\frac {2 b^{2} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {2 i b^{\frac {5}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {d \,b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 a^{\frac {3}{2}}}\) | \(356\) |
| default | \(-\frac {f \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{6 a \,x^{6}}+e \left (-\frac {\sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {2 b \sqrt {b \,x^{4}+a}}{21 a \,x^{3}}-\frac {2 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {b \left (b \,x^{4}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}+\frac {b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{16 a^{2}}\right )+c \left (-\frac {\sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {2 b \sqrt {b \,x^{4}+a}}{45 a \,x^{5}}+\frac {2 b^{2} \sqrt {b \,x^{4}+a}}{15 a^{2} x}-\frac {2 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\) | \(368\) |
-1/5040*(b*x^4+a)^(1/2)*(-672*b^2*c*x^8+840*a*b*f*x^7+480*a*b*e*x^6+315*a* b*d*x^5+224*a*b*c*x^4+840*a^2*f*x^3+720*a^2*e*x^2+630*a^2*d*x+560*a^2*c)/x ^9/a^2-1/840*b^2/a^2*(80*a*e/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2 )*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*( I/a^(1/2)*b^(1/2))^(1/2),I)+112*I*b^(1/2)*c*a^(1/2)/(I/a^(1/2)*b^(1/2))^(1 /2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4 +a)^(1/2)*(EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2) *b^(1/2))^(1/2),I))-105/2*a^(1/2)*d*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2 ))
Time = 0.17 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.49 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=\frac {1344 \, \sqrt {a} b^{2} c x^{9} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 315 \, \sqrt {a} b^{2} d x^{9} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 192 \, {\left (7 \, b^{2} c - 5 \, a b e\right )} \sqrt {a} x^{9} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (672 \, b^{2} c x^{8} - 840 \, a b f x^{7} - 480 \, a b e x^{6} - 315 \, a b d x^{5} - 224 \, a b c x^{4} - 840 \, a^{2} f x^{3} - 720 \, a^{2} e x^{2} - 630 \, a^{2} d x - 560 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{10080 \, a^{2} x^{9}} \]
1/10080*(1344*sqrt(a)*b^2*c*x^9*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1 /4)), -1) + 315*sqrt(a)*b^2*d*x^9*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 192*(7*b^2*c - 5*a*b*e)*sqrt(a)*x^9*(-b/a)^(3/4)*elliptic_f( arcsin(x*(-b/a)^(1/4)), -1) + 2*(672*b^2*c*x^8 - 840*a*b*f*x^7 - 480*a*b*e *x^6 - 315*a*b*d*x^5 - 224*a*b*c*x^4 - 840*a^2*f*x^3 - 720*a^2*e*x^2 - 630 *a^2*d*x - 560*a^2*c)*sqrt(b*x^4 + a))/(a^2*x^9)
Result contains complex when optimal does not.
Time = 4.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.58 \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} - \frac {a d}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 \sqrt {b} d}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} d}{16 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{6 a} + \frac {b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 a^{\frac {3}{2}}} \]
sqrt(a)*c*gamma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**4*exp_polar(I*pi)/ a)/(4*x**9*gamma(-5/4)) + sqrt(a)*e*gamma(-7/4)*hyper((-7/4, -1/2), (-3/4, ), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) - a*d/(8*sqrt(b)*x**10*s qrt(a/(b*x**4) + 1)) - 3*sqrt(b)*d/(16*x**6*sqrt(a/(b*x**4) + 1)) - sqrt(b )*f*sqrt(a/(b*x**4) + 1)/(6*x**4) - b**(3/2)*d/(16*a*x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(6*a) + b**2*d*asinh(sqrt(a)/(sqrt (b)*x**2))/(16*a**(3/2))
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}} \,d x } \]
\[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=\int { \frac {\sqrt {b x^{4} + a} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{10}} \,d x } \]
Timed out. \[ \int \frac {\left (c+d x+e x^2+f x^3\right ) \sqrt {a+b x^4}}{x^{10}} \, dx=\int \frac {\sqrt {b\,x^4+a}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{10}} \,d x \]